In [A], Arhangel'skii showed that for any T2
space X, |X|≤2
L(x)χ(x), where L(X) is the Lindelöf degree of X and χ(X) is the character of X.
In [B], Bell, Ginsburg and Woods improved this result, assuming normality, by showing that for T4 spaces X, |X|≤2
wL(x)χ(x), where wL(X) is the weak Lindelöf degree of X.
We introduce below a new cardinal function aL(X), the almost Lindelöf degree of X, which agrees with L(X) on T3 spaces, but which is often smaller than L(X) on T2 spaces, and show that for T2 spaces X,